Fitting Chord Into Circle

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Theorem

Into a given circle, it is possible to fit a chord equal to a given line segment which is not greater than the diameter of the circle.


In the words of Euclid:

Into a given circle to fit a straight line equal to a given straight line which is not greater than the diameter of the circle.

(The Elements: Book $\text{IV}$: Proposition $1$)


Construction

Euclid-IV-1.png

Let $ABC$ be the given circle and $D$ be the given line segment.

Let $BC$ be a diameter of circle $ABC$.

If $D = BC$ then the task is complete, as $BC$ is already fitted into $ABC$.


Otherwise, let $CE$ be cut off from $BC$ equal to $D$.

With center $C$ and radius $CE$, draw circle $EFG$.

Let $G$ be one of the points at which $EFG$ meets $ABC$.

Then $CG$ is the chord required.


Proof

As $C$ is the center of $EFG$, $CG = CE$ and so $CG = D$.

But $CG$ is a chord of $ABC$.

$\blacksquare$


Historical Note

This theorem is Proposition $1$ of Book $\text{IV}$ of Euclid's The Elements.


Sources